Explore the fourth lecture in a series on perturbation methods for nonlinear PDEs, delivered by Vishal Vasan at the International Centre for Theoretical Sciences. Delve into advanced topics such as weakly nonlinear shallow water-wave models, linear operators, and perturbation series. Examine the Boussinesq system, define inner products, and investigate the relationship between L+ and L operators. Engage with a Q&A session and study the damped KDV model. Gain insights into applying perturbation methods to complex nonlinear problems in cold atoms, water waves, and atmospheric science, with minimal prerequisites beyond familiarity with linear algebra.